TL;DR version: If one's reaction to the problem is "It is wrong to use expectation values, here," then one tends to find the problem trivial. If one's reaction to the problem is "Why can't one apply an expectation value, here?" then one has a much lengthier deliberation and discussion.Part of the issue that the mathematicians have with the problem appears to be that the problem is incompletely defined. Personally, I don't believe it is, because my reasoning ends at "why the heck would you use an expectation value in this case?" But, if we switch our perspective a bit, and assume (because the problem assumes) that an expectation value is OK to use, then it is fair to say that the problem doesn't give enough information to evaluate an expectation value.
Why? Because in the real world, there are upper and lower bounds to the amount of money that might realistically be put in the envelope, and potentially a probability distribution of particular amount pairs. E.g., if you found $0.01 in your envelope, you're not going to find 1/2 cent in the other, it will be 2 cents. (Or, there is a real USminted half cent in there, which is probably worth a great deal if you can find a coin collector to whom to sell it!) In general, if one assumes even a rudimentary guesswork "money spectrum", if one's envelope amount is on the low end, there is a good chance that the other envelope has more, and as one approaches the high end, switching envelopes gradually becomes more likely to "halve" the initial value you see.
Of course, it all depends on the distribution. If it's a flat distribution with a fixed maximum and minimum, e.g., $1  $20, no fractional dollars, then you have even more constraints that affect whether you should switch: any odd amount (which will always be < $10) will always be the lower of the pair, and any amount > $10 will always be the higher of the pair. If it's an even number of dollars less than or equal to 10, then, quite interestingly, the expectation value of switching really is 5/4 times the amount you see! Why? Because in this case we have 10 equally likely pairs, each with a 10% chance to come up. Looking at the value in the envelope you select then limits the possibilities down to only two pairs, each 50% likely. So, of the 20 cases (10 pairs, 2 possible choices for each) we know that switching doubles our value in 5 cases, halves our value in 5 cases, and has an expectation value of 5/4 our value in 10 cases. Note that this breaks the symmetry that the paradox implies, that switching back would also have a high expectation value, because one HAS to choose an envelope and open it to get the information needed to calculate the expectation value, and that expectation value isn't ALWAYS 5/4 the original value, but sometimes half or twice.
So, the mathematicians end up saying that if you assume a distribution that is any amount of money (unbounded, not necessarily integers of dollars or pennies), then you have an "improper prior", in which case an expectation value cannot be calculated. Or, stated more intuitively, if you know "how" the envelopes were filled, then looking at an envelope gives you enough information to determine whether you want to choose the other, and breaks the symmetry of using the same argument to switch. Lacking that knowledge of "how", the expectation value is undefined.
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03222012, 07:16 AM #11An argument is two people sharing their ignorance.
A discussion is two people sharing their understanding, even when they disagree.

03222012, 01:34 PM #12garbageGuest
When there's a 'paradox' in applying these sorts of mathematical concepts, it often means one of the following:
 The 'hammer' (e.g. expected value) that we use to nail down the problem can't actually applybecause of some minor technicality or assumption that we're not used to seeing when we approach problems using that hammer. Essentially, we're so focused on seeing nails, but we're actually looking at a screw. Or.. whatever, use an analogy more poignant than that.
 Related, and sometimes as a result of the above, we translate a word problem into a mathematical structure improperlyour definition of the problem is flippin' wrong.
 Our logical and/or mathematical structures are incomplete or inconsistent (something that we don't often accept). This is when we must build new, fancier tools, which often means approaching the problem in a new way, shifting our assumptions, etc.
I accepted your first answer as 'complete enough.' However, I did look into the paper, the Wikipedia article, and other resources out of curiositybut not because I felt that the detail was necessary. However however, my curiosity was driven by the fact that my master's studies focused on value and decision theory, and I find trying to frame decisions to be fascinating and useful. Hell, half of my career has been built around framing decisions, so I've got a vested interest in and curiosity about what's out there.
Ah, Ne
If we want to tie this to type, I would imagine that stereotypical Ti types would find the lynchpin, the Jenga block that stabilizes the whole structure, that knocks the whole argument in the OP's line of reasoning downand pull it. They would then dust off their hands, leave the scattered mess on the floor and say, "Welp, my work here is done," and then walk away.

03232012, 01:30 AM #13
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It seemed obvious to me that it was a matter of luck, even after knowing the content of one envelope.
I liked the 2nd answer better: ''1.5x expectancy''
The fact that I'm a poker fan probably made a difference.

03242012, 10:23 PM #14
I didn't look at the answers, but since this is about probability, isn't expectation value only used when you want to analyze the amount you would have if you grabbed the envelope X amount of times and want to know the average amount you would end up per grab, for a given number of X trials?
dsfjsdalgjsdf I am now Satan. Good day.

03242012, 10:27 PM #15
I don't know. Actually I don't get it. Because a coin flip is always 5050. If the envelopes are randomized each trial, then there's no way to assume that the average value of the grabs will converge anywhere specific in between A and 2A.
I guess. I don't know. I'm going to look at your answers now.

03242012, 10:55 PM #16
I am most satisfied with the second explanation, and it is the one that automatically occurred to me when I read the scenario description. The first seems imprecise; the third overelaborate.
I've been called a criminal, a terrorist, and a threat to the known universe. But everything you were told is a lie. The truth is, they've taken our freedom, our home, and our future. The time has come for all humanity to take a stand...

03252012, 06:37 AM #17
Yeah same, I just read up on it and basically came to something similar to the first solution, but that depends upon whether I understood it properly.
I also agree with @bologna about people's level of understanding being important to what explanation would work best; mine is limited and simple so a simple explanation is best.
So can I ask if I understood this correctly: The idea is to work out which envelope contains which amount of money and then be offered a switch, so then you need to work out whether you should switch based upon 'expectation' of what the other one might contain? I suppose there isn't really an answer, it's just a 50/50 chance, right?
Meh im not very intelligent so you might have to hand hold me through this one.

03252012, 08:20 AM #18
I like the second answer best, as it uses nice simple algebra rather than expecting me to know exactly how an expectation values is defined! I have more reasoning skills than knowledge, you know.
Don't make whine out of sour grapes.

03252012, 10:17 AM #19
The basic idea is that if you were given the situation where the 2nd amount was determined after you see the first amount, then the concept of "expectation value" applies. In simple terms, the expectation value is the "average" of all possible results. The expectation value of a fair roll of a 6sided die is "3.5", even though there is no 3.5 on the die, for example.
The expectation value is useful when that concept of average winnings actually means something. And it's more than just how likely you are to win. For instance, I could have a 1/3 chance to win and a 2/3 chance to lose, but if I only pay $3 to play, and get $12 if I win, then the average amount I'd win per play is 1/3(1x$12+3x(3)) = $1. Don't worry too much about the math, it depends a lot on specific wording. My point is that even though you're more likely to lose, on average, you still win, because the winnings when you win, on average, outstrip what you lose when you don't win.
Expectation value isn't foolproof. It is possible (google the St. Petersburg paradox) to have an infinite expectation value, but you'd never want to play because you'd run out of money before you ever won anything. (An infinite expectation value is why betting tables have a maximum possible bet, since you could just double your bet each time and eventually win, assuming you could borrow and bet any amount of money.) It is also possible, as in this thread's paradox, to simply misapply the math.
In the case of this paradox, you don't know what you're taking the average of in the first place. It's deliberately trying to trick you with fake reasoning by pretending that you're averaging doubling vs halving  the 1/2(A/2 + 2A) = 5/4 A is the average of doubling vs halving the value A  except you don't really know what A is. (A isn't the value in the envelope you see, it's that other number, X, that you see in the 2nd case, and you aren't really "doubling" or "halving" X.)
But yes, I suspect that the math aspect makes it hard to use for typing, because if one isn't comfortable with the math, one will go for the least mathematical explanation, regardless of type.An argument is two people sharing their ignorance.
A discussion is two people sharing their understanding, even when they disagree.

03252012, 10:32 AM #20
Yeah that does make a bit more sense, but as you said, the maths side of things is difficult for me and I cannot understand the maths at all. I suppose I should watch my brother play on a fruit machine and try to work it out from that /badjoke.
You've intrigued me now though, im probably going to spend hours looking over this topic and even trying to understand the maths behind it. I can't help it, if im ignorant of something I have no problem admitting it but I desire to understand it so as to broaden my understanding.
Ive never understood why people cant admit to their igorance on something. I suppose it has something to do with hubris, but to me it always causes more harm than good.
Anyhow thanks for the explanation and your patience.
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